Precalculus Solutions | Expert Assistance for Problems & Examples

Use condition $\sin θ - \cos θ = \sqrt{2}$
Please find the value of $\frac{1}{\sin^{10} θ} + \frac{1}{\cos^{10} θ}$

Ashleigh Mitchell 2022-04-08

Trigonometry problem $\frac{{\sin 100}^{\circ} + {\cos 70}^{\circ}}{{\cos 80}^{\circ} - {\cos 20}^{\circ}}$
I've done trigonometry in my earlier years of high school but I forgot a lot of rules. This is where I'm stuck on this problem:
$\frac{{\sin 100}^{\circ} + {\cos 70}^{\circ}}{{\cos 80}^{\circ} - {\cos 20}^{\circ}} =$
$\frac{\sin (90^{\circ} + 10^{\circ}) + \cos (60^{\circ} + 10^{\circ})}{\cos (90^{\circ} - 10^{\circ}) - \cos (30^{\circ} + 10^{\circ})} =$
$\frac{{\sin 90}^{\circ} {\cos 10}^{\circ} + {\cos 90}^{\circ} {\sin 10}^{\circ} + {\cos 60}^{\circ} {\cos 10}^{\circ} - {\sin 60}^{\circ} {\sin 10}^{\circ}}{{\cos 90}^{\circ} {\cos 10}^{\circ} + {\sin 90}^{\circ} {\sin 10}^{\circ} - {\cos 30}^{\circ} {\cos 10}^{\circ} + {\sin 30}^{\circ} {\sin 10}^{\circ}} =$
$\frac{{\cos 10}^{\circ} + \frac{1}{2} {\cos 10}^{\circ} - \frac{\sqrt{3}}{2} {\sin 10}^{\circ}}{{\sin 10}^{\circ} - \frac{\sqrt{3}}{2} {\cos 10}^{\circ} + \frac{1}{2} {\sin 10}^{\circ}} =$
$\frac{\frac{3}{2} {\cos 10}^{\circ} - \frac{\sqrt{3}}{2} {\sin 10}^{\circ}}{\frac{3}{2} {\sin 10}^{\circ} - \frac{\sqrt{3}}{2} {\cos 10}^{\circ}}$

zdebe5l8 2022-04-08

Trigonometry Identity:
$3 (\cot^{2} θ + 1) - \csc^{2} θ - 1 = \cot^{2} θ + \csc^{2} θ$

painter555ui8n 2022-04-08

Trigonometric Property when a+b+c=abc
then $\exists α, β, γ$ $a = \tan α; b = \tan β; c = \tan γ$ where $α + β + γ = π$
I don't know how to prove it

Malachi Mullins 2022-04-08

Trigonometric equations with 2 functions
Given the equation: $\sin^{2} x + \cos x = 0$

Briley Cabrera 2022-04-08

Tips on evaluating $lim_{x \to 0} \frac{\sqrt{1 - \cos (x^{2})}}{1 - \cos x}$ ?

Magdalena Warren 2022-04-08

The value of the expression $x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2$ when $x = \cot \frac{π}{12}$

Efrain Watkins 2022-04-07

For how many values of x in $0 \leq x < 2 π$ does $\frac{\sin (5 x)}{\sin (x)} - \frac{\cos (5 x)}{\cos (x)} = 2$
Using the sine and cosine addition rules, I tried to express $\sin (5 x)$ and $\cos (5 x)$ in terms of $\sin (x)$ and $\cos (x)$ . However, it took me quite a while, and I was wondering if there was a quicker way

Aedan Berry 2022-04-07

Fixing $γ, γ^{'}$ two reals numbers such that
$| γ^{'} | < \frac{1}{2} and 0 < γ < 1$ ,
and $α, β \in (0, π)$ , define
$x_{n} = \sin (2 π n γ + α + β) + \sin (2 π n γ^{'} + α - β)$
Could someone give me a simple statement equivalent to $x_{n} \geq 0$

Camila Glenn 2022-04-07

Finding $lim_{x \to 0} \frac{\sin (π \cos^{2} (\frac{x}{2}))}{\sin (\sin (x))}$
I have the following solution, only the first equality of which bothers me:
$lim_{x \to 0} \frac{\sin (π \cos^{2} (\frac{x}{2}))}{\sin (\sin (x))} = lim_{x \to 0} \frac{\sin (π \sin^{2} (\frac{x}{2}))}{\sin (\sin (x))} = lim_{x \to 0} π \frac{\sin (\frac{x}{2})}{2 \cos (\frac{x}{2})} . \frac{2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}{\sin (2 \sin (\frac{x}{2}) \cos (\frac{x}{2}))} . \frac{\sin (π \sin^{2} (\frac{x}{2}))}{π \sin^{2} (\frac{x}{2})} = 0$
What is the justification for replacing $\cos^{2}$ with $\sin^{2}$ ?

Quinn Dean 2022-04-07

Finding $\frac{1}{1 + {\tan 70}^{\circ}} + \frac{1}{1 + {\tan 20}^{\circ}}$

George Michael 2022-04-07

Find:
$lim_{x \to 0} \frac{1 - \cos 4 x}{x \tan 6 x}$

Rosa Townsend 2022-04-07

Find value of
$\prod_{r = 1} \sin (\frac{(2 r - 1) π}{2 n})$
Where $n \in N$ and $n > 1$

Aliyah Mendez 2022-04-07

find the x: $(\cos 3 x + \cos 4 x) (\cos 3 x + \cos x) = \frac{1}{4}$

Colt Rhodes 2022-04-07

Find the values of ${\sin 18}^{\circ}$

delitzo1d4 2022-04-07

Find the value of the following series.
$\tan θ + 2 \tan (2 θ) + 2^{2} \tan (2^{2} θ) + \dots + 2^{14} \tan (2^{14} θ) + 2^{15} \cot (2^{15} θ)$

Miguel Hanson 2022-04-07

Find the value of $10 \sin^{4} θ + 15 \cos^{4} θ = 6$

Boston Fitzgerald 2022-04-07

Find the value of $\cot (16) \cot (44) + \cot (44) \cot (76) - \cot (76) \cot (16)$

Aileen Rogers 2022-04-07

Find the value of $\cos (θ)$ given $\cot (θ)$
$23 \frac{\sqrt{530}}{530}$ but I keep getting $\cos (θ) = \frac{1}{\sqrt{530}}$ . Can someone please tell me what I’m doing wrong?
$1 + \cot^{2} (θ) = \frac{1}{\cos^{2} (θ)}$ // Pythagorean identity
$1 + 529 = \frac{1}{\cos^{2} (θ)}$ // Substitute $\cot^{2} (θ)$
$530 = \frac{1}{\cos^{2} (θ)}$
$530 \cos^{2} (θ) = 1$ // Divide both sides by $\cos^{2} (θ)$
$\cos^{2} (θ) = \frac{1}{530}$ // Divide both sides by 530
$\cos (θ) = \frac{1}{\sqrt{530}}$ // Square root both sides

Terrence Riddle 2022-04-07

Find the value of $\sin (\cos^{- 1} (\frac{1}{2}) + \sin^{- 1} (\frac{3}{5}))$

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